A396100 G.f. satisfies A(x) = (1/x)*A(A(x^2)) + A(A(A(x^2))).

1, 1, 2, 3, 6, 12, 17, 48, 52, 200, 162, 848, 502, 3631, 1605, 15730, 5063, 68501, 16314, 300199, 52203, 1320571, 169128, 5831378, 547153, 25825144, 1782944, 114682172, 5817259, 510476945, 19066119, 2277257478, 62627721, 10179548297, 206383073, 45590331780, 681724709, 204548718149, 2257730658

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..4000

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n has the following properties.

(1) A(x) = (1/x)*A(A(x^2)) + A(A(A(x^2))).

(2) A(A(x^2)) = x*(A(x) - A(-x))/2.

(3) A(A(A(x^2))) = (A(x) + A(-x))/2.

(4) A(A(x^2)) * A(A(A(x^2))) = x*(A(x)^2 - A(-x)^2)/4.

(5) A(x) converges for |x| < r where r = A(A(r^2)) at r = 0.460416184803533005220454820697720660656558236530176031947023...

(6) A(A(x)) converges for |x| < A(r^2) where A(r^2) = 0.286307088477904580502293384159028170286185128654277103146166...

(7) A(A(A(x))) converges for |x| < r^2 where r^2 = 0.211983063229041056608973065058713002287454641858749290265267...

(8) A(A(A(A(x)))) converges for |x| < q where A(q) = r^2 at q = 0.169686836320072198239498124401300712862395294961628283115207...

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 12*x^6 + 17*x^7 + 48*x^8 + 52*x^9 + 200*x^10 + 162*x^11 + 848*x^12 + ...
where A(x) = A(A(x^2))/x + A(A(A(x^2))).
RELATED SERIES.
The second and third iterations of A(x) begin
A(A(x)) = x + 2*x^2 + 6*x^3 + 17*x^4 + 52*x^5 + 162*x^6 + 502*x^7 + 1605*x^8 + 5063*x^9 + 16314*x^10 + ...
A(A(A(x))) = x + 3*x^2 + 12*x^3 + 48*x^4 + 200*x^5 + 848*x^6 + 3631*x^7 + 15730*x^8 + 68501*x^9 + 300199*x^10 + ...
the coefficients of which interleave to form this sequence.
SPECIFIC VALUES.
A(t) = 1 at t = 0.39673778500057382387495833993119539750836036095156...
A(A(t)) = 1 at t = 0.26255360314235395678135388941463883684788790226647...
A(A(A(t))) = 1 at t = 0.19926900486604179799565952253637071697535783657489...
A(t) = 2 at t = 0.44875853546126801043241091687812220875546168205825...
A(A(t)) = 2 at t = 0.28221464120396115535295360031881516608029031654276...
A(A(A(t))) = 2 at t = 0.20984343720647621630080412125020563105221221948719...
A(1/3) = 0.62229409951352684257488032885327282020663717653567...
A(1/4) = 0.36653400632354995092423995727875573563570860963597...
A(1/5) = 0.26388356638522249566069381720667867660165153315328...
A(A(1/9)) = 0.14797203450652118718881273999272077710797258706667...
A(A(A(1/9))) = 0.17837799599396328100844210887511048888271941533564...
  where A(1/3) = 3*A(A(1/9)) + A(A(A(1/9))).
A(A(1/16)) = 0.0720983262045680584895309647699304129073555208871...
A(A(A(1/16))) = 0.0781407015052777169661160981990340840062865260875...
  where A(1/4) = 4*A(A(1/16)) + A(A(A(1/16))).

Prog

(PARI) {a(n) = my(A = x, B, C, N=ceil(log(n)/log(2))); for(k=1, N,
A = truncate(A) + x*O(x^min(n, 2^k)); B = subst(A, x, A); C = subst(A, x, B);
A = subst(B, x, x^2)/x + subst(C, x, x^2); ); polcoef(GF=A +x*O(x^n), n)}
{upto(n) = a(n); Vec(GF)}
upto(40)

Crossrefs

Cf. A120566.

Sequence in context: A286906 A125867 A032727 * A323392 A174801 A324177

Adjacent sequences: A396097 A396098 A396099 * A396102 A396104 A396105

Keyword

nonn

Author

Paul D. Hanna, Jun 01 2026

Status

approved